42 Computational Modelling in Hydraulic and Coastal Engineering
as Couette flow. This flow is very important in engineering applications involving lubrication and friction between oscillatory machinery
components. In the preceding, the velocity field and the shear stress
at the oscillating plate are calculated. The solution is obtained under
the assumption that the lower plate moves periodically with a velocity
u
t U
t
T
lower ( )
sin
=
2π , while the upper remains still. The data used for this
example are as follows:
Maximum velocity of the lower plate = 1 cm/s
Period of oscillation = 500 s, 1000 s
Kinematic viscosity of the fluid = 0.01 cm 2 /s, 0.1 cm 2 /s
Distance between the plates = 19 cm
The governing equation of the physical phenomenon is the diffusion
equation (Equation 3.4), where the function f represents the velocity
distribution u(z, t). Thus the diffusion coefficient N is replaced by the
kinematic viscosity (ν) and the spatial step Δx by Δz (vertical axis).
The solution was accomplished by using the FTCS numerical scheme
(Equation 3.23). The initial condition is set as u(z, t = 0) = 0; and the
boundary conditions as u(z = 0, t) = 0 and u(z = H, t) = u up (t), where H
is the vertical distance between the plates. The expression used for estimating the shear stress is defined as τ
τ
ρ
ν
*
= =
du
dz
(Newtonian fluid).
The effects of the oscillation plate period and the fluid viscosity
on the shear stresses adjacent to the oscillating plate are shown in
Figure 3.10. It is evident that the shear stress increases with increasing
fluid viscosity.
Computer code 3.2
% Example 3.2 Two-Dimensional Couette Flow Caused by an
Oscillating Plate
% U = Velocity amplitude [cm/s];
% T = Oscillation period [s];
% v = Kinematic viscosity of the fluid [cm^2/s];
% Dt = Time step [s];
% Dz = Vertical step [cm];
% nm = Number of vertical discretization layers;
% tm = Number of time steps;
clc; clear all; close all;
% Input data;
U = 1;
T = 500;
T1 = 1000;
v = 0.01;
v1 = 0.1;